Author Archives: jgl

Q is not consonant: the Costantini-Watson argument

I have already given an argument for the non-consonance of the Sorgenfrey line Rℓ here. I would now like to explain why the space Q of rational numbers is not consonant either. That is quite a challenge. The most easily … Continue reading

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Compact scattered subsets and a topological game

Showing that Q is not consonant is quite an ordeal. I have finally managed to understand one of the existing proofs of this fact, due to Costantini and Watson. This would be a bit too long to cover entirely in … Continue reading

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Topological lattices with small semilattices

I would like to explain a clever counterexample due to Jimmie Lawson in 1970, or rather a slight variant of it, pertaining to the theory of topological semilattices and to a property that crops up naturally, namely having small semilattices. … Continue reading

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When do the upper (a.k.a., lower Vietoris) and Scott topologies coincide on the Hoare hyperspace of a space?

I would like to talk about a nifty, recent result due to Yu Chen, Hui Kou, and Zhenchao Lyu. There are two natural topologies on the Hoare hyperspace of a space X, the Scott and the lower Vietoris topology, and … Continue reading

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L-domains, stable open sets, and stable Stone duality

Stone duality relates topological spaces and locales (or frames). But there are really many sorts of Stone dualities. In 1997, Yixiang Chen studied Stone dualities that relate so-called L-domains to so-called distributive D-semilattices. This was refined later in a common … Continue reading

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Irredundant families, the Smyth powerdomain, the Lyu-Jia theorem, and the baby Groemer theorem

A ∩-semilattice of sets is a family of sets that is closed under finite intersections, and it is irredundant if and only if all its non-empty elements are irreducible. That sounds like a ridiculously overconstrained notion, but I will give … Continue reading

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Sheaves and streams II: sheafification, and stratified étale maps

In part I, I explained how one can build the étale space of a presheaf F over a topological space X. I will show how one can retrieve a sheaf from an étale map, leading to a nice adjunction and … Continue reading

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Sheaves and streams I: sheaves of locally monotone maps

Sheaves are a fundamental notion. In this post and later posts, I would like to explain some of the basic theory of the most mundane notion of sheaves: sheaves of sets over a topological space. My real goal is really … Continue reading

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Statures of Noetherian spaces I: maximal order types of wpos

I have had a very gifted masters 2 student from mid-May to late July, Bastien Laboureix. He mostly solved the questions I had left open here. I wanted to report on his work, but that is a lot too technical … Continue reading

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Topological functors I: definition, duality, limits and colimits

I have briefly mentioned topological functors in a recent post. It is time for me to explain what they are. This is a truly wonderful concept, which abstracts topological spaces away and concentrates on the key properties of the forgetful … Continue reading

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